Find the modulus and argument of z = -3 + 3i; express in polar form.

Find the modulus and argument of z = -3 + 3i; express in polar form.

• A. r = 3√2; θ = π/4; z = 3√2 (cos π/4 + i sin π/4)
• B. r = 3√2; θ = -π/4; z = 3√2 (cos -π/4 + i sin -π/4)
• C. r = 3√2; θ = 3π/4; z = 3√2 (cos 3π/4 + i sin 3π/4)
• D. r = 6; θ = 3π/4; z = 6 (cos 3π/4 + i sin 3π/4)

Answer: (C)

Finding the modulus and argument means measuring how far the number is from the origin and the angle its vector makes with the positive real axis. For z = -3 + 3i, the modulus is r = sqrt((-3)^2 + 3^2) = sqrt(9 + 9) = 3√2. The point (-3, 3) sits in the second quadrant, so the angle is between π/2 and π. The reference angle is arctan(|3/(-3)|) = arctan(1) = π/4, so the actual angle is θ = π − π/4 = 3π/4. Therefore z = r(cos θ + i sin θ) = 3√2 (cos 3π/4 + i sin 3π/4). This matches the option with r = 3√2 and θ = 3π/4; the other options either place the angle in a different quadrant or use the wrong modulus.

Finding the modulus and argument means measuring how far the number is from the origin and the angle its vector makes with the positive real axis. For z = -3 + 3i, the modulus is r = sqrt((-3)^2 + 3^2) = sqrt(9 + 9) = 3√2. The point (-3, 3) sits in the second quadrant, so the angle is between π/2 and π. The reference angle is arctan(|3/(-3)|) = arctan(1) = π/4, so the actual angle is θ = π − π/4 = 3π/4. Therefore z = r(cos θ + i sin θ) = 3√2 (cos 3π/4 + i sin 3π/4). This matches the option with r = 3√2 and θ = 3π/4; the other options either place the angle in a different quadrant or use the wrong modulus.